Developing Learners’ Fluency in Mathematics:
How Games Can Help
ARTICLE BY DREW POLLY, PH.D.
What is Fluency?
The current emphasis in schools on unfinished learning has led teachers and educational leaders to devote more time and energy to the concept of fluency; specifically basic fact fluency and procedural fluency. In both cases, fluency refers to learners demonstrating flexible strategies or thinking, efficient processes, and accuracy. Based on research and recommendations from mathematics education leaders, efficient strategies are not necessarily faster, but rather include fewer steps to reach the answer.
For example in terms of basic fact fluency, when adding 8 and 5, a learner may start at 8 and count on by 1s 5 times to reach 13, where another learner may start at 8 and break 5 into 2 and 3, and move 2 from 8 to reach 10 and move 3 from 10 to reach 13. The latter example is more efficient with 2 jumps compared to the prior example which had 5 jumps of 1.
In this blog we discuss how mathematics games can develop both basic fact fluency and procedural fluency.
Basic Fact Fluency
Basic fact fluency is commonly referred to as learners’ ability to use mental strategies and/or know from memory sums within 20 and their related subtraction relationships (e.g., 7 + 5, 5 + 7, 12 – 5, and 12 – 7), and/ or products of one-digit numbers and their related division relationships (e.g., 8 x 6, 6 x 8, 48 ÷ 6, and 48 ÷ 8). The table below gives an overview on the Basic Fact Fluency expectations in the Common Core State Standards.
Basic Fact Fluency
K.OA.5 Fluently add and subtract within 5.
1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.
2.OA.2 Fluently add and subtract within 20 using mental strategies. By the end of Grade 2, know from memory all sums of two one-digit numbers.
3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
Learners’ basic fact fluency is best developed when they are given opportunities to develop an understanding of strategies, practice using them, and consider when those strategies should be used.
For example, when multiplying 6 x 7, learners may think of the 7 as a 5 and a 2, and think of 6 x 7 as 6 groups of 5 joined with 6 groups of 2 to reach an answer of 42.
However, when multiplying 6 x 9, learners may think of 9 as the same as 10 take away 1 so they may mentally think about 6 x 9 being the same as 6 groups of 10 minus 6 to reach an answer of 54.
Mathematics educators have emphasized the need for learners to be flexible and knowledgeable about when it is most appropriate to use specific strategies.
When learners play math games that support basic fact fluency development, they have meaningful opportunities to apply strategies used to solve basic math facts embedded in the game. In many games, learners have agency to determine which strategy they can use and would be best suited to find a solution. For example, in the game Filling Water Balloons, learners pick a card (Figure 1) that has a total number of water balloons and the number of popped balloons. Learners have to determine how many balloons are left and then put that number of counters on their game board.
Learners may start at 2 and count on 3 numbers to reach 5, start at 5 and count backwards 2 to land on 3, or quickly show 5 fingers and take 2 away leaving 3. These strategies are pathways to helping learners demonstrate the accuracy, efficiency, and flexibility that accompanies basic fact fluency. A benefit of gameplay is that students explain their strategies and processes for finding the answer. These conversations help learners process their thinking, and refine their mathematical ideas as they communicate to peers.
Procedural fluency refers to the learners’ ability to flexibly, efficiently and accurately utilize mathematical procedures to solve problems, such as adding, subtracting, multiplying, or dividing multi-digit numbers, or simplifying expressions. The flexibility aspect of procedural fluency can be illustrated by the following student strategy selections.
For example, when multiplying 39 x 24 students may find the partial products of 30 x 20, 30 x 4, 9 x 20, and 9 x 4 and then add those 4 products up. Another student may use a doubling and halving strategy to make an equivalent problem that is easier to solve, such as doubling 39 and cutting 24 in half repeatedly.
NCTM advocates developing procedural fluency from conceptual understanding. In my example this means t learners need ample time and experiences making sense of the concepts of mathematical operations as well as place value before focusing more on procedural fluency. While students are developing conceptual understanding, math games provide a meaningful and supportive context for learners to deepen their understanding and start to grow in their procedural fluency.
Math games allow learners to choose which numbers and operations to use and encourage learners to estimate possible answers given the parameters of their options during game play. As an example, in the game Surf’s Up, learners get meaningful practice by selectively choosing which numbers they want to multiply in order to get a product that fits within a range of numbers (e.g., between 100 and 120, between 120 and 140). While playing, students must estimate, mentally compute products, and determine the possible products for their different options in their hand. In this game, specifically, conceptual understanding continues to be developed as students consider how to multiply a two-digit number by a one-digit number in various ways. This may include decomposing the two-digit number into a multiple of ten and the ones-digit, such as, 23 x 7 may be solved by adding the products of 20 x 7 and 3 x 7). Additionally, learners may choose to decompose the one-digit factor into 5 and whatever is left, specifically multiplying 23 x 5 and then 23 x 2 to find the answer.
Game play promotes mathematical dialogue, strategic thinking, and fluency development as students engage in enjoyable practice.
As you reflect on basic fact fluency and procedural fluency, consider how math games may support learners’ development. Questions you may ask yourself could include:
- What skills or concepts could learners benefit from with meaningful practice embedded in math games?
- Where in your math period could you make room for students to play math games?
- As students are playing math games, what questions or supports can you provide to help students process and communicate their mathematical ideas?